Singular factorizations, self-adjoint extensions, and applications to quantum many-body physics

TL;DR

This paper explores how factorizing differential operators can introduce singular interactions in quantum models, leading to new exactly solvable many-body systems of Calogero-Sutherland type.

Contribution

It presents a novel method to construct exactly solvable quantum many-body systems using factorizations and self-adjoint extensions, applicable to models with singular interactions.

Findings

Explicit solutions for quantum models with singular interactions

A new method for constructing Calogero-Sutherland type systems

Extension of single-particle factorizations to many-body cases

Abstract

We study self-adjoint operators defined by factorizing second order differential operators in first order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum mechanical models like the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero-Sutherland type.